Deutschherren Stuben: On the left
Directly across from the Grüner Hof is another traditional restaurant, the Deutschherren Stuben.
They do in fact get somewhat less sun on this side of the Market Square, but it's still a very pleasant place to sit and have a meal. I forget what I ate here (it was a while ago), but I do remember I was very satisfied.
By the way there are about two dozen restaurants in Weikersheim, so you should have no trouble getting a meal even when the town is full of opera goers.
Grüner Hof: Since 1804
This venerable restaurant is located on the market square (on the sunny side of the market square, as they point out on their website), right beside the entrance to Weikersheim Castle. If the weather is nice, you can eat outside on the square, otherwise the inside is very pleasant, too.
I've eaten here twice so far, and was very satisfied, also met some nice people here even in an even-numbered (non-opera) summer.
They claim to speak German, English, Russian, Swedish and Spanish. (I can confirm that their German is fine, didn't try any of the others.)
They also have a hotel with thirty beds.